Computing optimal mean/downside risk frontiers: the role of ellipticity 185
If we assume, without loss of generality that μ 1 μ 2 , when there is no shortselling allowed, then μ 1 μ (^) p μ 2 for 0 ω 1. We shall concentrate on
this part of the frontier, but could consider extensions for ω 1 or ω * 0.
Now define θτp^2 () as the lower partial moment of degree 2 with truncation
point τ. Then,
θτ τ
τ
ppr pdf r drpp
(^22) ()( ) ( )
∫∞
(8.28)
However , an alternative representation in terms of the joint pdf of r 1 and r 2
is available. Namely,
θτ τ^22 pp()( r)pdf r r dr dr(12 1 2, )
∫ℜ
(8.29)
where ℜ ()rr 12 ,|r 1 ωωτr 2 ( ).^1
∗∗−
{}^
We can now change variables from ( r 1 , r 2 ) to ( r 1 , r p ) by the (linear)
transformations
rrpωω∗∗ 1211 1( )rand rr
(8.30)
Therefore ,
dr dr 12 dr dr 1 p
1
1
01
()ω
∗ if ω∗
(8.31)
If ω 1, then the transformation is ( r 1 , r 2 ) ( r 2 , r p ).
Now ,
θτ
τ
ω
ω
ω
τ
p
rpp
pdf r
rr
(^2) dr
2
1
1
11
()
()
()
∞ ∗
∗
∫ ∗
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
, 11 drp
∞
∞
∫
(8.32)
This equation gives us the mean/semivariance locus for any joint pdf , i.e.,
pdf ( r 1 , r 2 ). As μ (^) p changes, ω changes and so does θτp^2 (). In certain cases, i.e.,
ellipticity, we can compute explicitly pdf ( r p ) and we can directly use Equation
(8.28). In general, however, we have to resort to Equation (8.32) for our
calculations.
In what follows, we present two results: First, for N 2 under normal-
ity, where Equation (8.28) can be applied directly and a closed form solution